Polarizability tensor interaction-induced

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

Here a designates the trace of the polarizability tensor of one molecule (l/47i o) times the factor of a represents the electric fieldstrength of the quadrupole moment q2. Other non-vanishing multipole moments, for example, octopoles (e.g., of tetrahedral molecules), hexadecapoles (of linear molecules), etc., will similarly interact with the trace or anisotropy of the polarizability of the collisional partner and give rise to further multipole-induced dipole components. 

On a scale of the order of atomic size, molecular multipole fields vary strongly with orientation and separation. As a consequence, one will generally find induced dipole components arising from field gradients of first and higher order which interact with the so-called dipole-multipole polarizability tensor components, such as the A and E tensors. 

Besides the isotropic overlap-induced dipole component familiar from the rare gas pairs, we will now in general have other significant induced dipole components if molecules are present, namely multipole-induced and distorted frame-induced dipole components, see Chapter 4 for details. Moreover, these anisotropic dipole components couple with the polarizability tensor and thus give rise to simultaneous transitions in two (or perhaps more) molecules. Furthermore, molecules in general interact with more or less anisotropic forces which to some extent does also affect the spectra of molecular systems. 

In Ref. (316) the SAPT approach has been applied to compute the interaction-induced polarizability of the helium diatom. Before discussing the Raman spectra obtained from the SAPT collision-induced polarizabilities, let us discuss the importance of various physical contributions to the parallel and perpendicular components of the collision-induced polarizability tensor. 

The induced component arises from the interaction of the molecular polarizability tensors, 7t, with electric fields. The induced dipole moment on molecule i is given by ... 

We mention that, by contrast, the more familiar Raman spectra arising from the permanent polarizabilities of the individual (noninteracting) molecules of the complex are not considered a part of the supermolecular spectra, or of CILS. In ordinary Raman spectroscopy of rarefied gases the invariants of the permanent molecular polarizability tensor are conveniently considered to be not affected by intermolecular interactions, an approximation that is often justified because induced spectral components are usually much weaker than ordinary allowed Raman bands. 

In general, the polarizability is a tensor whose invariants, trace and anisotropy, give rise to polarized and fully depolarized light scattering, respectively. Collision-induced light scattering is caused by the excess polarizability of a collisional pair (or a larger complex of atoms or molecules) that arises from the intermolecular interactions. In Section I.l, we are concerned with the definition, measurement, and computation of interaction-induced polarizabilities and their invariants. 

Thus the polarizability tensor of the molecule is written as a sum of local atomic polarizabilities, each modified through dipolar interactions with the electric dipole moments on all the other atoms induced by the electric vector of the incident light wave. Similarly for the local atomic polarizabilities appearing in the origin-dependent parts of the optical activity tensors. But unlike the bond polarizability development, no allowance can be made for intrinsic local optical activity tensors Gj ap and Aj since these now pertain to spherical atoms. We refer to the original articles for the explicit Raman intensity and optical activity expressions generated by the atom dipole interaction theory. 

The matrix p is a combined set of the external electrostatic fields that represent the effects of the QM field on the EFP polarizability tensors and the PCM potential, while w is a combined set of induced dipoles and surface charges. The physical meaning of the supermatrix equation (3) is that the EFP induced dipoles and PCM induced charges are uniquely determined by the external field and potential therefore, the right hand side of Eq. (3) involves only the external field /potential, and the left side involves only the induced EFP dipoles and PCM charges. The interactions among the induced dipoles and charges are implicitly described with the matrix B. The supermatrix Eq. (3) can be solved either with direct inversion or various iterative methods. 

The polarizability tensor of these free oriented interacting molecules depends on the Euler angles d, 62,(f> = (f> — intermolecular distance R, and the internuclear distances in the molecules r and ra. For considered model the total induced dipole moment of atomic system is written as... 

As it was noted in Chaps. 3 and 4 the dipole moment modulus and the polarizability tensor invariants practically do not change for the family of the most stable configurations [52]. We can expect, that the first-hyperpolarizability tensor invariants which are important for description of interaction-induced hyper-Rayleigh scattering, also change weakly for all most stable configurations. 

In practice, the calculation involves three steps the calculation of the property for three systems A- B, A and B. If the desired goal is the calculation of the interaction-induced mean dipole polarizability a) or second dipole hyperpolarizability (y), one must first obtain the Cartesian components of both tensors for all three systems. The general definition of the mean is... 

In our study, the polarizability tensor of methane calculated from Gaussian 94 (22) is isotropic within the numerical accuracy of the calculation. Therefore the formula and calculations could be simplified. However, for most guest molecules in clathrate hydrates that are not as symmetric as methane, this derivation gives a general approach to characterize the induced dipole-dipole interaction. 

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